In certain games it is impossible to compute expected utility; take voting case for the most fundamental example. Due to exogenous rights every eligible agent has the same voting power. Moreover election cannot depend on intensity of the voters for it would favour candidates with fanatic electorate and, what is a worse flaw, it is not strategy proof.

Hence voting should rely on the idea of ordinal preference relation - complete and transitive relation $R$. $xR_iy$ means that for the $i$th agent alternative $x$ is at lest as good as $y$.

\subsection{Failure of plurality voting}

Plurality voting is the essential part of classical democracy. Making decision through plurality voting means to constitute an alternative voted by the majority of voters. It is suitable provided having two alternatives. Then indeed any alternative chosen by majority of voters is a rational choice of entire community.

But we encounter an impediment when any other alternative comes into view. Consider voting over three alternatives, e.g. three candidates for president: A, B and C; preferred by 60\%, 40\% and 0\% of voters respectively. Majority choice is apparent. Nevertheless, if we take into account entire preference relation of voters, obviousness of that selection may be refuted.

Let's peruse the following example. Voters divide into two subsets of identical preferences:

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\% of voters & $1^{ST}$ choice & $2^{ND}$ choice & $3^{RD}$ choice \\
\hline
\hline
60\% & A & B & C \\
40\% & B & C & A \\
\hline
%\caption{Electorate distribution}
%\label{tab:votex1}
\end{tabular}
\end{center}

Naturally, candidate A is still plurality winner, but for 40\% of the voters she is the worst choice. Reasonable decision in this case would be electing candidate B as a president, who is at least second choice for every voter.

\subsection{Borda count}

Approach presented in previous section was introduced and formalized by Jean-Charles de Borda in 1770 and it is now known as \textit{Borda count}. Borda's idea relies on very simple rule, namely scoring zero points for the last place in personal ranking of every voter, one point for second worst, etc. The winner of \textit{Borda count} is the candidate with the highest total score.

Having $p$ candidates, general formula computing single score of candidate $X$ being on $k$-th position in voter's personal ranking is:
\begin{equation}
S_X=p-k
\end{equation}

\subsubsection{Example}
Let's check the scores of candidates from previous section's example assuming 100 voters:
\begin{align*}
S_A = 60(p-1)+40(p-3)&=120 \\
S_B = 60(p-2)+40(p-1)&=140 \\
S_C = 60(p-3)+40(p-2)&=40
\end{align*}

Hence, the Borda winner is candidate B, which is plausible to every voter in considered example.

\subsubsection{Disadvantages}
Nevertheless, Borda count has one significant flaw, namely it fails Independence if Irrelevant Alternatives rule (IIA). To show it, let's remove candidate C from the voting. Then Borda score changes in such way:
\begin{align*}
S_A = 60(p-1)+40(p-2)&=60 \\
S_B = 60(p-2)+40(p-1)&=40 
\end{align*}

Removal of the candidate who has little support may yield diametrical changes in final result, which contradicts IIA rule.  

\subsection{Scoring methods}

\textit{Borda count} is remarkable method belonging to the family of scoring methods. This vast family includes, as well, plurality voting, which simply assigns one point to every top ranked alternative and zero points to other choices. Another noteworthy member of the family is antiplurality, which scores 1 point to every choice apart from the bottom ranked candidates. The antiplurality winner is the least unwanted alternative. 

\subsection{Condorcet method}

Another voting method which is superior than usual plurality is \textit{Condorcet method}. It requires entire ranking of candidates from every voter but it considers only duels between every two alternatives. Let's examine the results of \textit{Condorcet method} in given example:
\begin{itemize}
\item A prevails B 60:40
\item A prevails C 100:0
\item B prevails C 100:0
\end{itemize}
Thus outcome of the \textit{Condorcet method} coincides with plurality voting result. How then this method is claimed to be superior than plurality? Although the result is the same, \textit{Condorcet method} takes into account entire preference relation instead of top ranked alternatives.

However giving less plausible result than \textit{Borda count}, \textit{Condorcet method} has one significant advantage, i.e. it fulfills IIA rule by reason of heeding only duels regardless of how many alternatives separate considered candidates.

Anyhow, Condorcet himself noticed that his method has one major defect, namely in some conditions the ultimate preference relation computed from the duels may be cyclic, which goes against the requirement of transitivity. Proposed solution of ignoring the "weakest link", i.e. disregarding the pairs of alternatives with the lowest difference, has lead to the famous \textit{reunion paradox}.

\subsection{Preference aggregation}

The core concept of voting methods is to set up a preference aggregation which will equitably incorporate preference profiles of the agents. Resulting aggregation is anthropomorphic; it treats a group of voters as a single agent.
\\
$A$ - set of outcomes \\
$N$ - set of agents \\
$F$ - aggregation method of individual preferences \\
$R_i$ - $i$th agent's preference ordering \\
$\tilde{R}$ - profile of the voting community \\

Ordinal collective welfare is expressed as follows: 
\begin{equation}
F: \tilde{R}=(R_i, i\in N) \longmapsto R^*
\end{equation}

\subsubsection{Scoring methods}

\begin{equation}
\begin{split}
&R^s=F(\tilde{R}): \\
&\forall_{x,y\in A}xR^sy\Longleftrightarrow 
\sum_{i\in N}S_x(R_i)\geq\sum_{i\in N}S_y(R_i)
\end{split}
\end{equation}

\subsubsection{Condorcet method}

\begin{equation}
\begin{split}
&R^c=F(\tilde{R}): \\
&\forall_{x,y\in A}\,xR^cy\Longleftrightarrow 
|\{i\in N:\,xP_iy\}|\geq|\{i\in N:\,yP_ix\}| \\
&where\,\,xP_iy\Longleftrightarrow (xR_iy\,\wedge\,\neg yR_ix)
\end{split}
\end{equation}

